AN APPLICATION OF LATTICE THEORY TO QUASIGROUPS M. F. SMILEY The purpose of this note is to show that O. Ore's general formula­ tion of the Jordan-Holder theorems in partially ordered sets1 [3] yields the Jordan-Holder theorems for loops [l] which have recently been obtained by A. A. Albert [2]. We shall assume that the reader is familiar with the papers [l, 2, 3] of O. Ore and of A. A. Albert. We begin with an examination of certain congruence relations in loops. We are led to characterize the normal divisors of A. A. Albert as those subloops which commute and associate with the elements of the loop. Our proof of the fundamental quadrilateral condition of O. Ore [3] is then based on this characterization. A loop ® is a quasigroup with an identity element e. For each non- null subset §C® we define a relation Hp on ®® as follows: (1) If x, y G ®, then xHpy in case y G x$£>. THEOREM 1. The relation Hp is a congruence relation for the loop ® if and only if (i) x£(xh)$, (ii) (x$)(y$)C(xy)$ for every x, yG® and &G§. PROOF. Let Hp be a congruence relation for ®. For each #G® and hÇL§ we have xHp(xh) by the definition of Hp. Since Hp is symmetric we have (xh)Hpx, x(E(xh)fQ. If also ^G®, AiG^S then yHp(hyi), and since Hp preserves multiplication we obtain (xy)Hp(xh)(hyi)1 (xh)(yhi) G(#30§. Conversely, let § satisfy (i) and (ii). If #G®, choose feG§ and we have #G(#^)^pC#§ by (i) and (ii). Thus Hp is reflexive. If also yG® and xHpy, we have y=xfa, and (i) yields #G(#&i)§=:y§, yHpx. Thus Hp is symmetric. If also sG® and yHpz, we have z — yh^ = (^i)feG^^ by (ii). Thus Hp is transitive and is an equivalence relation. The relations xHpy, zHpw yield (xz)Hp(yw) by (ii) and we conclude that Hp is a congruence relation for ®. Remark 1. If R is a congruence relation for a loop ®, then the sub­ set $K = [x ; xRe ] is a subloop of ®. For clearly e G9Î and 9Î is closed with Presented to the Society, April 29, 1944; received by the editors April 8, 1944. The opinions and assertions contained in this paper are the private ones of the author and are not to be construed as official or reflecting the views of the United States Navy Department or of the naval service at large. 1 Numbers enclosed in brackets denote the references given at the end of the paper. 782 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use AN APPLICATION OF LATTICE THEORY TO QUASIGROUPS 783 respect to multiplication. If we have ri# = f2, then eRfi, xRx give xR(rix)y xRr2, r2Re, xRe, #G3t Similarly xri = f2 implies #G9Î and 9Î is a subloop of ®. With R = HP, we obtain 9? = |) and thus (i) and (ii) imply that § is a subloop of ©. (For groups ®, every congruence rela­ tion R is of the form Hp for suitable $C®0 Remark 2. We note the following consequences of (i) and (ii): (2) x(y$) C (xy)$, (3) (*$)y C (*y)©, (4) *($y) C (*y)$, (5) $x C x§. For finite § each of these inequalities2 is an equality. To see this for (2), note that for fixed x, y G® we have a single-valued mapping A—»&* defined by #(yft) = (xy)h*. This mapping is one-to-one since #(yfti) =#(yfe) implies Ai = &2. For a finite ^), we conclude that the set of images [fe*] = §, and consequently that (xy)&=x(y$). The same reasoning applies to the inequalities (3), (4), and (5). It is now easy to see that if § is a. finite subloop of ®, then Hp is a congruence rela­ tion for ® if and only if the set ^p commutes and associates with the elements of ® in the sense that (6) xQ = $x, (7) (xy)$ = x(y$), (x$)y = x($y), (§x)y = $(*?) for every x, yG®. THEOREM 2. -4 subloop $ of a loop ® satisfies (6) <md (7) i/ awd <wZy if § is a normal divisor of ® in the sense of A. A. Albert. PROOF. Consider a subloop § of a loop ® which satisfies (6) and (7). The permutation group §p generated by the right multiplication of ® gives rise to the cosets X(QP of Albert. We first show that the cosets x&p=xJQ. Clearly X{QPZ)X!Q. The elements of $p are finite products 1 l of permutations of the form Rh and Rjr for AG§. If y = xRjr , then yh =x,y=y(hhx) = {yh)h2 = xh2, y =xRh2. It follows that each element of x&p may be written in the form xhÇzJX$&* We next note that (8) (*$)(?$) = (*y)$ is an immediate consequence of (6) and (7). To see that the cosets x& form a loop we now consider the equations 2 This result is due to G. N. Garrison. See §4 of his fundamental paper, Quasi- groups, Ann. of Math. vol. 41 (1940) pp. 474-487. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 784 M. F. SMILEY [October (9) (*©)(«>$) = y$, (*©(*©) = y©. Solutions of (9) are provided by the solutions of xw = y and zx=*y for w and 2. The uniqueness of the solutions w$ and z& of (9) is readily verified. We have proved that ®/$ is a loop and hence that © is a normal divisor of © in the sense of A. A. Albert. To prove the converse it suffices to use the representation ^> = eT furnished by Albert's Theorem 3 [l ]. Using this theorem, the verifica­ tions of (6) and (7) are quite simple and we shall omit them. Remark 3. The equations (6) and (7) may be replaced by (10) (x$)y = x(y$), (11) (xy)$ = x(y$) for every x, 3>G©. To see this, set x = e in (10) to obtain (6). THEOREM 3. The intersection & of a system ($«; a G Q) of normal divisors of a loop © is a normal divisor of ®. PROOF. Clearly § is a subloop of ®. We shall prove §x&§ for every #G®. If AG§, aG 0, we have hx=xha for some haG&a- The left cancellation law then shows that the set [ha] aGfi] is singular and its sole member is an element of ^p. The remainder of the proof consists in a repetition of this argument which proves (6) and (7). COROLLARY. The set of all normal divisors of a loop ®, when ordered by set inclusion, forms a complete lattice. PROOF. It suffices to remark that ® is a normal divisor of ®. We shall use KJ and C\ to denote the lattice operations of this lattice. THEOREM 4. If & and $ are normal divisors of a loop ®, then §$ is a normal divisor of © and ^p$ = §U$* PROOF. Clearly eG§$ and §$ is closed with respect to multiplica­ tion. Now if x(hk)=hiki, we use (7) to get (xh)k2 = hiki. Two applica­ tions of (i) and use of (6) and (7) then give x = hzfo. We compute *($&) = (*€>)$ = (#*)$ = €>(#$) = &($tx) = ($$t)x. Thus #$ is a sub- loop of © satisfying (6). We omit the simple verification of (7). Since every normal divisor of © which contains § and $ must contain §$, we conclude that £>$ = §U$. COROLLARY. The lattice of normal divisors of a loop © is modular. THEOREM 5 (QUADRILATERAL CONDITION). If § and $1 are maximal normal divisors of a loop ®, then SQC\$ is a maximal normal divisor of the loop #, and ®/$£Ë£/£>n$. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use i944l AN APPLICATION OF LATTICE THEORY TO QUASIGROUPS 785 PROOF. Suppose on the contrary that 8 is a normal divisor of the loop £ such that ^)>8>^>n$. We shall prove that 8$ is a normal divisor of ®. We note that ® = $$ and to verify (10) we consider the equation [(A*)(Ai*i)](*i*«) = (**)[(M*)(Ms)] which we may write, using (8), (i), and (7) for $, as (12) (**i)*« = [h{h2hz)]h. Now if hi G8, we may use (10) for 8 to get h(h2l) = [h(h2hz) \k±. We now use (7) for $, cancel A, use (7) for $ again and cancel h2 to obtain l = hzfa. It follows that hÇi&, and hence that &öG^pn$C8. Conse­ quently A3G8 and we have proved [#(8$)];yC#[:y(8$)] for every x, y£z®. On the other hand, if A3G8, we may use (10) for 8 to get (hh)h2=[(hl)h2]ki. We now use (6) and (7) for $ and cancel h2} use (6) and (7) for $ and cancel h to obtain fa = lkz from which A1G8 follows as before. This completes the verification of (10). We apply the same reasoning to verify (11). We write [(A*)(Mi)](W = (**)[(*i*i)(M.)] as (hhi)h2= [h(hihz)]h* Then if A2G8, we have h(hj) =h[(hihz)k5] and we easily obtain A3G8. Also, if A3G8, we have {hh\}h2 = {hhi){lk^), h2 = lfa, A2G8. Thus (11) holds for 8$. To show that 8$ is a subloop of ® it now suffices to note that eG8$, (8$) (8$) =8$, and to prove that if (hk) Qiki) = l2k2y then h G8. But we may write this equation as hl\ = l2ki and get h = h(hh) = (hh)h = hfo. As before, &5G8, AG8. We have proved that 8$ is a normal divisor of ®. Note also that ®>8$>$, since if AG§ and A does not belong to 8, then hk = lki implies h = lk2, &2G8, AG8, a contradiction, and 8>^n$.